Donald Trump and his advisers have stood firm after a weekend of outrage over his vague and chaotically enforced ban on travel from seven Muslim-majority countries.

On Sunday afternoon, while attorneys argued with customs and border officials over the fate of people still detained at airports around the country, the US president released a statement that insisted on the legality – and non-religious premise – of his orders to temporarily halt the admission of refugees and ban some travel.

“To be clear, this is not a Muslim ban, as the media is falsely reporting,” Trump said.

During his campaign, Trump promised a “complete and total shutdown of Muslims entering the United States”. On Friday he said he would prioritize Christian refugees. In his statement on Sunday, the president compared his order to a 2011 decision by Barack Obama to delay visas for Iraqis, and said Obama’s White House had chosen the seven countries singled out in his order.

“This is not about religion – this is about terror and keeping our country safe,” Trump said, pledging to issue visas again “once we are sure we have reviewed and implemented the most secure policies over the next 90 days”.

He also expressed “tremendous feeling” for men, women and children fleeing civil war in Syria and said: “I will find ways to help all those who are suffering.” Trump’s order imposed an indefinite freeze on refugee admissions from the country.

Trump did not acknowledge the federal judges who in emergency hearings on Saturday night blocked parts of his orders and opened the door to a battle over the president’s constitutional powers.

Earlier on Sunday, two prominent Republican senators broke with the president over his order, warning that it betrayed American values by turning away refugees and green-card holders.

US travel ban - a brief guide

The executive order signed by Donald Trump suspends the entire US refugee admissions system, already one of the most rigorous in the world, for 120 days. It also suspends the Syrian refugee program indefinitely, and bans entry to the US to people from seven majority-Muslim countries – Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen – for 90 days. The order has prompted a series of legal challenges, while thousands of Americans have protested outside airports and courthouses in solidarity with Muslims and migrants.

“Such a hasty process risks harmful results,” John McCain and Lindsey Graham said in a joint statement. “We fear this executive order will become a self-inflicted wound in the fight against terrorism.”

In an interview with CBS’s Face the Nation, McCain said that the order “in some areas will give Isis some more propaganda”.

Shortly after releasing his official statement, Trump used Twitter to denigrate the senators, calling them “wrong” and “sadly weak on immigration”.

“Senators should focus their energies on ISIS, illegal immigration and border security instead of always looking to start World War III,” the president tweeted.

He had earlier insisted that only draconian measures could protect the US from outside tumult. “Our country needs strong borders and extreme vetting, NOW,” he tweeted. “Look what is happening all over Europe and, indeed, the world – a horrible mess!”

Attorneys, meanwhile, told reporters “rogue” border agents around the country were still detaining people or trying to deport them. Large protests at major airports that began on Saturday carried into Sunday, and in Washington hundreds of protesters marched on the White House and the Capitol, chanting “let them in” and “Paul Ryan sucks”.

Ryan, the House speaker, has not commented on the court orders to stop deportations, or the defiance of some customs and border officials. A spokeswoman said on Saturday that he supported the order and does not consider it a religious test.

Trump’s White House advisers defended the president’s orders, which suspended the US refugee program for 120 days, ended the Syrian refugee program indefinitely, and halted travel from seven countries – cutting off legal residents from their families and jobs.

Appearing on NBC’s Meet the Press, White House chief of staff Reince Priebus said 325,000 travelers had entered the US on Saturday and that 109 were detained.

“Most of those people were moved out,” he said. “We’ve got a couple dozen more that remain and I would suspect that as long as they’re not awful people that they will move through before another half a day today.”

In an abrupt, apparent change from the White House’s original policy, Priebus said the order would no longer affect green card holders. He then contradicted himself, and suggested that “other countries” may be added to the travel ban.

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“Maybe some of those people should be detained,” he said, although valid visa holders have already passed through an arduous screening and interview process. Priebus told NBC the administration would “apologize for nothing”.

In Brooklyn on Saturday night, federal judge Ann Donnelly had observed that the people detained would have been allowed into the US without incident, had they arrived two days earlier.

Similar orders from federal judges in Virginia and Massachusetts followedand Democrats rallied to help detained people and protesters, with Congressman John Lewis going to the Atlanta airport and New York mayor Bill de Blasio telling CNN’s State of the Union the order “violates our constitutional norms”.

The Department of Homeland Security responded to the judge’s ruling on Saturday morning, saying agents would continue to enforce Trump’s orders.

“No foreign national in a foreign land, without ties to the United States, has any unfettered right to demand entry into the United States or to demand immigration benefits in the United States,” the agency said. Only in the fourth paragraph of its reply did the DHS concede that officials “will comply with judicial orders”.

Another close Trump adviser, Kellyanne Conway, insisted that the president’s order had not disrupted US domestic and foreign policy, caused problems at airports, or provoked scorn from European allies. Conway falsely told Fox News Sunday the decision of the federal judge in New York “doesn’t really affect the executive order at all”.

She then said that problems for “1%” of travellers was “a small price to pay” for security. “This whole idea that they’re being separate and ripped from their family, it’s temporary.”

A handful of Republicans in Congress broke with Trump on the order, at least partially, harkening back to when the businessman originally described it as a “complete and total shutdown” on Muslim migration and was criticized by many more people in his party.

Only a few spoke out on Saturday, including senators Jeff Flake and Ben Sasse, who respectively called the order “unacceptable” for legal residents and “too broad”.

The Republican leader in the Senate, Mitch McConnell, said he would not completely defend or criticize the White House’s order.

“I am opposed to religious tests,” he told ABC’s This Week. “It’s going to be decided in the courts as to whether or not this has gone too far.”

Trump’s order is being challenged as a violation to the constitution’s guarantees of due process and against religious discrimination, meaning that its intent could prove an important factor in deciding its legality. His adviser Rudy Giuliani has said he was asked to design “the right way to do it legally”.